Partial Differential Equations (PDEs) are central to both pure and applied mathematics. Any quantity which changes in space and time will satisfy certain partial differential equations because the laws of nature, and the laws of physics specifically, relate a quantity to its rate of change in space and time.
Prof. Ugur Abdulla leads the Analysis and Partial Differential Equations Unit at the Okinawa Institute of Science and Technology (OIST). The unit's goal is to reveal and analyze the mathematical principles reflecting natural phenomena expressed by PDEs.
"The temperature distribution in this room satisfies a well-known partial differential equation called the heat equation or diffusion equation. This equation mathematically describes how heat spreads through a medium over time, capturing the essence of temperature changes in our surroundings," Prof. Abdulla explained. "Technologies like power generation, electric motors, and wireless communication are modelled by coupled PDEs, called Maxwell equations. In the general theory of relativity, Einstein field equations form a system of non-linear PDEs which relate the spacetime geometry to the distribution of mass-energy, momentum, and stress."
Prof. Abdulla, an Azerbaijani-American, was born and raised in Baku, Azerbaijan, then part of the Soviet Union. He lived on a street named after Nobel-winning physicist and fellow Baku native Lev Landau. From a young age, he excelled in math, physics, and sport, becoming a national Math Olympiad winner and freestyle wrestling champion. He graduated high school with distinction and an academic gold medal - awarded to the top student in a graduating class. At 15, Prof. Abdulla began studying math at Baku State University, following in Landau's footsteps. By 21, he earned his Ph.D. from the USSR Academy of Sciences and became a full professor at 27. His career took him to England, Germany, and the United States, where he held prestigious academic positions. In 2022, he joined OIST, drawn by its unique research and educational environment.
Solving Kolmogorov's problem while swimming
In 2008, Prof. Abdulla solved a longstanding open mathematical problem posed by Andrey Kolmogorov in 1928, by discovering how spacetime geometry at infinity relates to the formation of the singularity in diffusion processes and then proving a mathematical rule that could test for the predictability of the process. He vividly recounted the moment he solved Kolmogorov's problem after years of dedicated work; the most significant breakthrough in his distinguished research career: "I felt as though I stood on the shoulders of two giants - the great Soviet mathematician Andrey Kolmogorov, and the great American mathematician Norbert Wiener. I remember exactly when I solved the problem. I was swimming and it just hit me. Instead of rushing out of the pool, I kept swimming to enjoy the unforgettable moment."
Dr. Daniel Tietz, a researcher in the unit, is building on Prof. Abdulla's research: "I'm expanding on the new discoveries that Prof. Abdulla made on the heat equation by looking at more general situations. I'm studying how heat moves through anisotropic materials - substances whose properties change depending on the direction in which they are measured. Anisotropic materials have different conductivity properties in different directions. I'd like to understand how these properties affect heat flow and how this flow can be described mathematically."
Prof. Abdulla discussed how solving major mathematical problems often demands innovative approaches and careful validation by other mathematicians. His solution to the Kolmogorov problem led to insights in singularities. A mathematical singularity refers to points where mathematical calculations break down or become infinite. In many cases, these singularities represent critical natural phenomena, such as black holes in space and turbulent fluid flows. His recent paper provides a comprehensive way to understand singularities and could help resolve some of the biggest unsolved problems in math and science.
Early detection of cancer using PDEs
The unit is undertaking an exciting project to tackle one of society's major health challenges, cancer. Early detection of cancer is crucial for effective treatment. Prof. Abdulla and his collaborators in the USA have developed a new method for early detection of cancerous tumors based on electrical impedance tomography - a safe and non-invasive imaging method - and optimal control theory of partial differential equations. This project is personal for him, after losing a close relative to an aggressive form of cancer.
This technique sends a small electrical current through the body and the voltage is measured. The method is based on the fact that the electrical conductivity of cancerous tumors is twice that of healthy tissue. By taking electrical measurements from multiple angles and analyzing them using PDEs, scientists can collect enough data to create detailed images that reveal potential cancer locations.
The tricky part is understanding what is inside the body from measurements taken on small portions of the body surface. This is called an "inverse problem" and is a major challenge for mathematicians and computer scientists because it requires solving complex mathematical problems in an unstable setting. This new method overcomes these difficulties.
Dr. Jose Rodrigues, a unit researcher who developed a computational framework for the cancer detection method, added: "We've recently created computer code to test Prof. Abdulla's new method through numerical experiments. The results show that this technique provides a powerful way to identify cancer development regions with high resolution. The next step is to communicate with doctors and medical institutions to test this technology on real medical data and then apply it to detect early cancer development in patients."
Vision for a global mathematical center at OIST
Prof. Abdulla joined OIST with the goal of making it one of the world's leading centers in mathematics. "We want OIST to be a key place for sharing math knowledge globally," he said. "Our work with partner institutes helps us connect with different parts of the world, sharing knowledge and providing opportunities for PhD students to work on cutting edge problems in the frontline of mathematics."
An example of how the unit links global math communities is the OIST-Oxford-SLMath Summer School 2024, hosted from July 29 to August 9 at OIST. The collaborative event brought together 40 PhD students from 35 universities worldwide, including 5 Japanese institutions. The program featured two mini courses led by Prof. Abdulla and Prof. Gui-Qiang G. Chen from the University of Oxford. The courses covered various aspects of PDEs and gave participants a broad understanding of current challenges and methods in the field.
Dr. María Sánchez-Muñiz, a lecturer at The City College of New York, attended the summer school with her infant daughter. She investigates how asymmetric melting could help explain observed phenomena like explosive methane releases in thawing permafrost regions. "My research is on developing conceptual models using PDEs to analyze the thermodynamics of permafrost thaw. My work examines how heat diffusion through soil layers - modeled through parabolic PDEs - captures the complex phase transitions that occur as permafrost responds to rising global temperatures."
Ayrton Pablo Almada Jimenez, a PhD student at the University of Arizona, highlighted his interest in the course: "I'm working on power grid simulations and analysis of rare events involving power grids, both distribution and transmission levels. Most of the analysis that we do requires solving partial differential equations, so I'm enjoying reviewing a lot of the fundamental principles and learning new techniques to solving classic PDE problems."
Together with colleagues from MIT, Oxford, UC Berkeley, the University of Arizona, and the University of Alabama, Prof. Abdulla is planning to organize a Semester Program at SLMath on "Singularities & PDEs Representing Natural Phenomena". The program will bring together experts from four key areas in analysis & PDEs: potential theory, nonlinear hyperbolic systems of conservation laws, nonlinear Schrodinger equations, and fluid mechanics. It promises to be a leading forum advancing knowledge in the field, training PhD students and postdoctoral scientists, and inspiring collaboration among researchers.
